
\input graphicx
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%\vsize=1.05\vsize



\def\UseTimesRoman{
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\def\BR{\Bbb R}             % Besondere Buchstaben
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\def\BQ{\Bbb Q}
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\input amssym.def            % small letters for UNIX,  not: AMSsym.def
\input epsf.def% \input epsf %for UNIX
%\input epsf          %\input epsf.def for MAC f"ur BILDER!!
\input pics.tex 

\input BoxedEPS
\SetTexturesEPSFSpecial
\HideDisplacementBoxes

\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\Lf{\vskip1pt\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum
\def\R90{{\rm Rot}(90^\circ)}
\def\Dd#1{{\partial \over \partial #1}}

\nopagenumbers

\vglue 10pt



\cl {\bf About Spherical Ellipses}
\cl { See ATO for Planar Ellipses}
%\cl{\sc Elementary Definition, Practical Application}
\Lf
In {\sc 3DXplorMath} the {\it Default Morph} shows a family of
ellipses with fixed focal points $F_1,F_2$ as the larger axis $aa$ varies from its
allowed minimum $e=bb/2$ to its allowed maximum $\pi-e=\pi-bb/2$. Another
interesting morph is $0.11\le aa \le 1.43,\ \ 0.2\le bb \le \pi-0.2$: the distance of 
the focal points increases until they are almost antipodal and the major axis is
only slightly longer than the distance of the focal points.
\vskip1mm\noindent\hrule \vskip1mm\
\lf
{\sc Elementary Definition.} 
Many elementary constructions from planar Euclidean geometry have natural
analogues on the twodimensional sphere $\BS^2$. For example, we can take
the definition of planar ellipses and use it on the sphere as follows:
Pick  
two points $F_1,F_2 \in\BS^2$ of spherical distance $2e:=dist(F_1,F_2) < \pi$ 
and define the set of points $P\in\BS^2$ {\it for which the sum of the
distances  to the two points $F_1,\ F_2$ equals a constant $=:2a$}, i.e. the
set:
$$ \eqalign{
\{P\in\BS^2;\ dist(P,F_1)+dist(P,F_2) = 2a\}, \cr
\hbox{to be a \ {\sc Spherical Ellipse}.\hskip0.45in}
}$$
In the Euclidean plane there is only one restriction between the
parameters of an ellipse: $ 2e < 2a $. Since distances on $\BS^2$
cannot be larger than $\pi$ we have two restrictions in spherical
geometry: $ 2e < 2a < 2\pi-2e $. \goodbreak\noindent
For fixed focal points, i.e. for fixed $e$, these curves cover the
sphere (we allow that the smallest and the largest ellipse degenerate to
great circle segments). One observes that the ellipse with $2a = \pi$
is a great circle and that ellipses with $2a > \pi$ are congruent to
ellipses with $2a < \pi$ and focal points $-F_1,-F_2$. \lf
 This is because $dist(P,F) = \pi - dist(P,-F)$ implies \lf
$$\eqalign{\pi< 2a = dist(P,F_1) + dist(P,F_2) \Rightarrow \hskip5mm\cr 
dist(P,-F_1) + dist(P,-F_2) = 2\pi-2a < \pi.
}$$
Similarly, on the sphere one does not need to distinguish between
ellipses and hyperbolas:
$$ \eqalign{
&\{P\in\BS^2;\ dist(P,F_1)+dist(P,F_2) = 2a\} = \cr
&\{P\in\BS^2;\ dist(P,F_1)-dist(P,-F_2) = 2a-\pi\} . 
}$$
\lf
{\sc Practical Application.}
These curves are used since more than 50 years in the {\sc LORAN} System
to determine the position of a ship on the ocean as follows. Consider a
pair of radio stations which broadcast synchronized signals. If one
measures at any point $P$ on the earth the time difference with which a
pair of signals from the two stations arrives, then one knows the
difference of the two distances from $P$ to the radio stations. Therefore
sea charts were prepared which show the curves of constant difference
of the distances to the two radio stations. This has to be done for
several pairs of radio stations. In araes of the ocean where the families
of curves (for at least two pairs of radio stations) intersect reasonably
transversal it is sufficient to measure two time differences, then a look
on the sea chart will show the ship's position as the intersection point
of two curves, two sperical hyperbolas. On the site \lf
http://webhome.idirect.com/...    \lf
\Tilde jproc/hyperbolic/index.html    \hskip1cm or\lf
\Tilde jproc/hyperbolic/lorc\_hyperbola.jpg     \lf
this is explained by the following map:
\vfil\eject

\vglue 50pt
\includegraphics[width=2.75in]{loran.pdf}
\vfil\eject

\cl
{\sc Elementary Construction, 3DXM-demo}
\Lf
Begin by drawing a circle of radius $2a$ around $F_1$ (called
{\it Leitkreis} in German). Next, for every point $C$ on this circle
we find a point $X$ on the spherical ellipse as follows: \lf
Let $M$ be the midpoint of the great circle segment from $C$ to $F_2$
and let $T$ be the great circle through $M$ and perpendicular to that
segment. In other words, $T$ is the symmetry line between $C$ and $F_2$.
Finally we intersect $T$ with the {\it Leitkreis} radius from
$F_1$ to $C$ in $X$. --- Because we used the symmetry line $T$ we have
$dist(X,C) = dist(X,F_2)$ and therefore:
$$ \eqalign{
dist(X,F_1)+dist(X,F_2) &= dist(X,F_1)+dist(X,C) \cr
&= dist(C,F_1) = 2a.
}$$
It is easy to prove that the great circle $T$ is tangent to the ellipse
at the point$X$.
\LF

\cl{\sc Connection with Elliptic Functions}
\Lf
We met a family of ellipses all having the same focal points ('confocal')
and also the orthogonal family of confocal hyperbolas in the visualization
of the complex function $z \to z + 1/z$. In the same way two orthogonal
families of confocal spherical ellipses show up in the visualization of
elliptic functions from {\it rectangular tori} to the Riemann sphere
(choose in the Action Menu: {\it Show Image on Riemann Sphere} and in the
View Menu: {\it Anaglyph Stereo Vision}). --- Note that in the plane
all such families of confocal ellipses and hyperbolas are essentially
the same, they differ only in scale. On the sphere we get different
families for different rectangular tori, i.e. for different quadrupels
of focal points $\{F_1,F_2,-F_1,-F_2\}$.
\LF

\cl{\sc An Equation for the Spherical Ellipse}
\Lf
Abbreviate $\alpha:=dist(X,F_1),\ \beta:=dist(X,F_2)$. The definition of
a spherical ellipse says:
$$ \eqalign{
&\cos(2a) = \cos(\alpha+\beta) =
\cos\alpha\cos\beta-\sin\alpha\sin\beta.\cr
&\hbox{with }
\cos\alpha = \langle X,F_1\rangle,\ \cos\beta = \langle X,F_2\rangle.
}$$
We want to write the equation in terms of the scalar products which
are linear in $X$. Therefore we replace $\sin^2 = 1-\cos^2$ to get:
$$ \eqalign{
&(1-\cos^2\alpha)(1-\cos^2\beta) =
(\cos\alpha\cos\beta-\cos(2a))^2
\cr
&\hbox{or} \cr
&1-\cos^2\alpha-\cos^2\beta =
-2\cos(2a)\cos\alpha\cos\beta+\cos^2(2a)
\cr
&\hbox{or, by replacing the cosines by the scalar products:} 
\cr &\hskip1cm
\sin^2(2a)\langle X,X\rangle - 
                  \langle X,F_1\rangle^2 - \langle X,F_2\rangle^2 =
\cr &\hskip1.4cm
-2\cos(2a)\cdot \langle X,F_1\rangle\cdot \langle X,F_2\rangle.
}$$
Observe that this is a homogenous quadratic equation in $X=(x,y,z)$. In
other words: Our spherical ellipse is the intersection of the unit sphere
with a quadratic cone whose vertex is at the midpoint of the sphere. So we
get the surprisingly simple result: If one projects a spherical ellipse
from the midpoint of the sphere onto some plane then one obtains a
(planar) conic section.

\medskip
\leftline{H.K.}


\bye